Ferkans — Interactive Telecom Tutor

The RIS as a Multi-User Interference Sculptor

Single-user RIS is an SNR booster: every phase choice helps the one user, no one else. Multi-user RIS is a dual-purpose device: phases can simultaneously amplify one user's signal and deliberately misalign another's. The RIS thus reshapes not just the channel strengths but the interference structure of the system — choosing who can be served simultaneously and who must time-share. This is the topic of the chapter.

The golden thread is explicit: by programming $\boldsymbol{\Phi}$ , the RIS simultaneously steers beams toward desired users and nulls them at undesired ones. Active beamforming spatially multiplexes; passive beamforming shapes the channel so that the active multiplexing becomes easier. Together, they let small MU-MIMO arrays serve more users with less mutual interference.

MU-RIS

A multi-user MIMO system where a base station with $N_t$ antennas serves $K$ users jointly, aided by one or more reconfigurable intelligent surfaces. The RIS shapes inter-user interference as well as desired signals, enabling simultaneous beam focusing toward multiple users through a single panel.

Related: Sum Rate, Max Min Fairness

SOCP (Second-Order Cone Program)

A convex optimization problem with second-order cone constraints of the form $\|\mathbf{A}\mathbf{x} + \mathbf{b}\| \leq \mathbf{c}^T \mathbf{x} + d$ . Polynomial-time solvable. Max-min rate problems in RIS reduce to bisection over a sequence of SOCPs, one per feasibility check.

Definition:
Multi-User MISO-RIS Downlink

A BS with $N_t$ antennas serves $K$ single-antenna users via a single $N$ -element RIS. The BS transmits $\mathbf{x} = \sum_{k=1}^{K} \mathbf{v}_{k} s_k$ , with $s_k \sim \mathcal{CN}(0, 1)$ i.i.d. and $\mathbf{v}_{k} \in \mathbb{C}^{N_t}$ . The total transmit power is $\text{tr}(\mathbf{W}^{H} \mathbf{W}) \leq P_t$ .

The effective channel for user $k$ is $\mathbf{h}_{k,\text{eff}}^H = \mathbf{h}_{k,d}^H + \mathbf{h}_{k,2}^H \boldsymbol{\Phi} \mathbf{H}_1$ , yielding received signal

$y_k = \mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k} s_k + \sum_{j \neq k} \mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j} s_j + w_k, \qquad w_k \sim \mathcal{CN}(0, \sigma^2).$

The per-user SINR is

$\text{SINR}_k(\mathbf{W}, \boldsymbol{\Phi}) = \frac{|\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k}|^2}{\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 + \sigma^2}.$

The inter-user interference $\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2$ is the new ingredient relative to single-user. The RIS can reduce it (by making users' effective channels more orthogonal) or increase it (a bad local optimum), depending on whether the optimization steers wisely.

,

Theorem: Achievable Rate Region with RIS

Let $\mathcal{R}_{\text{MU}}(\boldsymbol{\Phi})$ denote the achievable rate region of the MU-MISO downlink with effective channels $\{\mathbf{h}_{k,\text{eff}}(\boldsymbol{\Phi})\}$ and power $P_t$ . The RIS-aided achievable rate region is

$\mathcal{R}_{\text{RIS}} = \bigcup_{\boldsymbol{\phi}: |\phi_n| = 1} \mathcal{R}_{\text{MU}}(\boldsymbol{\Phi}).$

$\mathcal{R}_{\text{RIS}}$ is not convex in general (the union of convex regions is not convex), but its achievable boundary is monotone-improving in the sense that $\mathcal{R}_{\text{MU}}(\boldsymbol{\Phi} = \mathbf{0}) \subseteq \mathcal{R}_{\text{RIS}}$ (the direct-channel region is a special case of the RIS-aided region, recovered by choosing $\boldsymbol{\Phi}$ to cancel the reflected path).

For a fixed $\boldsymbol{\Phi}$ , we face a standard MU-MISO downlink with effective channels $\{\mathbf{h}_{k,\text{eff}}\}$ . Its achievable rate region is known — a union of rate tuples over feasible precoders with total power $\leq P_t$ . The RIS expands this region because every $\boldsymbol{\Phi}$ gives a different set of effective channels, each with its own achievable region.

Proof

Fix $\boldsymbol{\Phi}$

For each $\boldsymbol{\Phi}$ , the system is a standard MU-MISO downlink with known effective channels, whose achievable region $\mathcal{R}_{\text{MU}}(\boldsymbol{\Phi})$ is characterized by Costa's dirty-paper coding result or approximated by linear-precoding methods.

Union over $\boldsymbol{\Phi}$

Every feasible $\boldsymbol{\Phi}$ contributes its rate region. The RIS-aided region is the union, which is not generally convex — two $\boldsymbol{\Phi}$ values at the boundary of their respective regions need not have a convex combination on the overall boundary.

Inclusion

Choosing $\boldsymbol{\phi}$ to make $\mathbf{h}_{k,2}^H \boldsymbol{\Phi} \mathbf{H}_1 \mathbf{v}_{k} = 0$ (e.g., by matched-filtering the direct path and aligning the RIS phases to cancel) recovers the direct-only region. Hence inclusion. $\blacksquare$

The Rank Constraint, Multi-User Edition

From Chapter 3, the cascaded channel $\mathbf{H}_2 \boldsymbol{\Phi} \mathbf{H}_1$ has rank $\leq \min(\text{rank}(\mathbf{H}_1), N, \text{rank}(\mathbf{H}_2))$ . For multi-user pure-LoS scenarios where both hops are rank 1, the cascaded channel is rank 1 in total — regardless of $K$ ! All users share a single spatial direction from the BS's perspective.

Multi-user multiplexing through a pure-LoS RIS requires user- specific RIS-UE paths to be linearly independent. If UEs differ in angle (common for geographically separated UEs), the RIS-UE channels $\{\mathbf{h}_{k,2}\}$ span a $K$ -dimensional subspace and multiplexing becomes possible. In a severe shared-path case, the RIS devolves to a time-division server with no multiplexing gain.

Rate Region With and Without RIS

For a 2-user MISO system, plot the achievable rate region $(R_1, R_2)$ with and without the RIS. The RIS version expands the region by reshaping the effective channels; the $N^2$ per-user gain translates to a larger outer boundary. Increase inter-user correlation to see the RIS advantage shrink (users in the same spatial direction cannot be multiplexed, RIS or not).

Parameters

RIS elements

N

64

BS antennas

N_t

4

Transmit SNR (dB)10

User channel correlation0.3

Key Takeaway

The multi-user RIS problem is fundamentally about interference shaping. For a fixed active beamformer, the RIS phases determine the inter-user interference pattern. Choosing $\boldsymbol{\Phi}$ to make $\mathbf{h}_{k,\text{eff}}$ orthogonal across users drastically reduces interference and boosts the sum rate; choosing it to concentrate power on the weakest user raises the max-min rate. These are fundamentally different objectives that require different optimization machinery, developed below.

Common Mistake: Don't Assume Equal Power Across Users

Mistake:

"In MU-RIS with $K$ users, just split power $P / K$ per user and optimize $\boldsymbol{\Phi}$ ."

Correction:

Equal power is strictly suboptimal for sum-rate maximization. Water-filling gives more power to stronger users; the RIS can amplify strong users further by matched-filter phases, making the water-filling gap even larger. For max-min fairness, equal power is closer to optimum — but still suboptimal because the RIS should favor the weakest user's channel. Always let the active-beamformer subproblem optimize the power split.

Why This Matters: Standard MU-MIMO Precoding with an Extra Knob

Everything you know from MU-MIMO precoding — WMMSE sum-rate, max-min SOCP, block-diagonalization — carries over to MU-RIS with one modification: the channels are now functions of $\boldsymbol{\Phi}$ , itself optimized in an outer loop. AO neatly separates the two concerns: inner MU-MIMO precoding in each step, outer RIS update between steps. Your existing MU-MIMO codebase becomes a MU-RIS codebase by wrapping it in an outer AO loop over $\boldsymbol{\Phi}$ .

See full treatment in Multi-User Multibeam Operation

The Multi-User RIS Signal Model